Difference between revisions of "Diels-Alder Reaction Experimental Design"

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\begin{array}{rcl}
 
\begin{array}{rcl}
 
\dot{x}(t) &=& f(x(t), u(t), p)  
 
\dot{x}(t) &=& f(x(t), u(t), p)  
 +
\end{array}
 +
</math>
 +
 +
== Constraints ==
 +
 +
The control variables are constrained with respect to the mass of sample weights (initial mass):
 +
 +
 +
<math>
 +
\begin{array}{cll}
 +
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 
 +
\end{array}
 +
</math>
 +
 +
and to the mass of active ingredient content (fraction of active substances):
 +
 +
<p>
 +
<math>
 +
\begin{array}{cll}
 +
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2  \ + \ n_{a4} \ \cdot \ M_4 }  \le 0.7
 
\end{array}  
 
\end{array}  
 
</math>
 
</math>
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The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:
 
The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:
  
−
<p>
+
 
 
<math>
 
<math>
 
\begin{array}{cll}
 
\begin{array}{cll}
−
  \displaystyle \min_{x, G, F, Tc, n_{a1}} && trace(F^{-1} (t_{end})) \\[1.5ex]
+
  \displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]
 
  \mbox{s.t.} \\
 
  \mbox{s.t.} \\
−
\dot{x}(t) & = & f(x(t), u(t),p),  \\
+
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p),  \\
 
\\
 
\\
 
  h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
 
  h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
 
  \\
 
  \\
−
  \dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
+
  \dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\
 
  \\
 
  \\
−
  \dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
+
  \dot{F}(t) & = & \sum\limits_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\
 
  \\
 
  \\
 
  0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4    \\
 
  0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4    \\
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  0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2  \ + \ n_{a4} \ \cdot \ M_4 }  \\
 
  0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2  \ + \ n_{a4} \ \cdot \ M_4 }  \\
 
  \\
 
  \\
−
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
+
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2]   \\  
−
\\
+
                                      \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273  & t \in [2,8]   \\
−
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273  , \quad \forall \, t \in [2,8] \\
+
                                      \vartheta_{up} + 273 &  t \in [8,t_{end}]
−
\\
+
                    \end{array} \right. \\
−
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
+
& & x  \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\
−
\\
+
\dot{z}^i(t)  & = & w^i(t)  \\
−
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.
+
z(0) & = & 0 \\
 +
w^i(t) &\in& [0,1] \\
 +
0 &  \le & 4 - z^i(t_f). \\
 
\end{array}  
 
\end{array}  
 
</math>
 
</math>
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|1
 
|1
 
|}
 
|}
 +
 +
Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.
  
 
{| class="wikitable"
 
{| class="wikitable"
−
|+Parameters
+
|+Fixed parameters
 
|-
 
|-
 
|Name
 
|Name
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{| class="wikitable"
 
{| class="wikitable"
−
|+Control variables
+
|+Optimization/control variables
 
|-
 
|-
 
|Name
 
|Name
 
|Symbol
 
|Symbol
 
|Interval
 
|Interval
 +
|Initial value Exp 1
 +
|Initial value Exp 2
 +
|Initial value Exp 3
 +
|Initial value Exp 4
 
|-
 
|-
 
|Initial molar number 1
 
|Initial molar number 1
 
|<math>n_{a1}</math>
 
|<math>n_{a1}</math>
−
|[0.4,9.0]
+
|[0,10.0]
 +
|1.0
 +
|1.0
 +
|1.0
 +
|1.0
 
|-
 
|-
 
|Initial molar number 2
 
|Initial molar number 2
 
|<math>n_{a2}</math>
 
|<math>n_{a2}</math>
−
|[0.4,9.0]
+
|[0,10.0]
 +
|1.0
 +
|1.0
 +
|1.0
 +
|1.0
 
|-
 
|-
 
|Initial molar number 4
 
|Initial molar number 4
 
|<math>n_{a4}</math>
 
|<math>n_{a4}</math>
 
|[0.4,9.0]
 
|[0.4,9.0]
 +
|2.0
 +
|2.0
 +
|2.0
 +
|2.0
 
|-
 
|-
 
|Concentration of the catalyst
 
|Concentration of the catalyst
 
|<math>c_{kat}</math>
 
|<math>c_{kat}</math>
−
|[0.0,6.0]
+
|[0,10.0]
 +
|0.0
 +
|1.0
 +
|2.0
 +
|3.0
 +
|}
 +
 
 +
{| class="wikitable"
 +
|+Control function
 +
|-
 +
|Name
 +
|Symbol
 +
|Time interval
 +
|Value interval
 +
|Initial value Exp 1
 +
|Initial value Exp 2
 +
|Initial value Exp 3
 +
|Initial value Exp 4
 +
|-
 +
|Initial molar number 1
 +
|<math>\vartheta(t)</math>
 +
|<math>[t_0,2]</math>
 +
|[20.0,100.0]
 +
|20.0
 +
|60.0
 +
|40.0
 +
|20.0
 +
|-
 +
|Initial molar number 1
 +
|<math>\vartheta(t)</math>
 +
|<math>[2,8]</math>
 +
|[20.0,100.0]
 +
|20.0
 +
|60.0
 +
|40.0
 +
|20.0
 
|-
 
|-
 
|Initial molar number 1
 
|Initial molar number 1
 
|<math>\vartheta(t)</math>
 
|<math>\vartheta(t)</math>
 +
|<math>[8,t_{end}]</math>
 
|[20.0,100.0]
 
|[20.0,100.0]
 +
|20.0
 +
|60.0
 +
|40.0
 +
|20.0
 
|}
 
|}
  
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</math>
 
</math>
  
−
== Constraints ==
+
== Source Code ==
  
−
The control variables are constrained with respect to the mass of sample weights:
+
* The VPLAN code using [[:Category: VPLAN | VPLAN]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]
−
 
+
−
 
+
−
<math>
+
−
\begin{array}{cll}
+
−
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 
+
−
\end{array}
+
−
</math>
+
−
 
+
−
and to the mass of active ingredient content:
+
−
 
+
−
<p>
+
−
<math>
+
−
\begin{array}{cll}
+
−
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2  \ + \ n_{a4} \ \cdot \ M_4 }  \le 0.7
+
−
\end{array}
+
−
</math>
+
  
 
== References ==
 
== References ==
Line 238: Line 302:
  
  
−
[[Category:VPLAN]]
 
 
[[Category:Optimum Experimental Design]]
 
[[Category:Optimum Experimental Design]]
 
[[Category:ODE model]]
 
[[Category:ODE model]]

Latest revision as of 15:17, 29 July 2016

The Diels-Alder Reaction is an organic chemical reaction. A conjugated diene and a substituted alkene react and form a substituted cyclohexene system. Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.


Model Formulation

The reactionkinetics can be modelled by the following differential equation system:

\begin{array}{rcl} \dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\ & & \\ \dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\ & & \\ \dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\ & & \\ \dot{n_4}(t) &=& 0 \end{array}

The reaction velocity constant k consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )

Total mass:

m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4

Temperature in Kelvin:

T(t) = \vartheta (t) + 273

The ODE system is summarized to:

\begin{array}{rcl} \dot{x}(t) &=& f(x(t), u(t), p) \end{array}

Constraints

The control variables are constrained with respect to the mass of sample weights (initial mass):


\begin{array}{cll} 0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10 \end{array}

and to the mass of active ingredient content (fraction of active substances):

\begin{array}{cll} 0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7 \end{array}

Optimum Experimental Design Problem

The aim is to compute an optimal experimental design \xi = (q,w) which minimizes the uncertainties of the parameters k_1, k_{cat}, E_1, E_{cat}, \lambda. So, we have to solve the following optimum experimental design problem:


\begin{array}{cll} \displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex] \mbox{s.t.} \\ \dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\ \\ h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\ \\ \dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\ \\ \dot{F}(t) & = & \sum\limits_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\ \\ 0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\ \\ 10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\ \\ 0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\ \\ 0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\ \\ \vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \end{array} \right. \\ & & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\ \dot{z}^i(t) & = & w^i(t) \\ z(0) & = & 0 \\ w^i(t) &\in& [0,1] \\ 0 & \le & 4 - z^i(t_f). \\ \end{array}


State variables
Name Symbol Initial value (t_0)
Molar number 1 n_1(t) n_1(t_0) = n_{a1}
Molar number 2 n_2(t) n_2(t_0) = n_{a2}
Molar number 3 n_3(t) n_3(t_0) = 0
Solvent n_4(t) n_4(t_0) = n_{a4}


Constants
Name Symbol Value
Molar Mass M_1 0.1362
Molar Mass M_2 0.09806
Molar Mass M_3 0.23426
Molar Mass M_4 0.236
Universal gas constant R 8.314
Reference temperature T_{ref} 293
St.dev of measurement error \sigma 1

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

Fixed parameters
Name Symbol Value
Steric factor k_1 p_1 \cdot 0.01
Steric factor k_{kat} p_2 \cdot 0.10
Activation energie E_1 p_3 \cdot 60000
Activation energie E_{kat} p_4 \cdot 40000
Catalyst deactivation coefficient \lambda p_5 \cdot 0.25

with p_j = 1, \ j =1, \dots, 5

Optimization/control variables
Name Symbol Interval Initial value Exp 1 Initial value Exp 2 Initial value Exp 3 Initial value Exp 4
Initial molar number 1 n_{a1} [0,10.0] 1.0 1.0 1.0 1.0
Initial molar number 2 n_{a2} [0,10.0] 1.0 1.0 1.0 1.0
Initial molar number 4 n_{a4} [0.4,9.0] 2.0 2.0 2.0 2.0
Concentration of the catalyst c_{kat} [0,10.0] 0.0 1.0 2.0 3.0
Control function
Name Symbol Time interval Value interval Initial value Exp 1 Initial value Exp 2 Initial value Exp 3 Initial value Exp 4
Initial molar number 1 \vartheta(t) [t_0,2] [20.0,100.0] 20.0 60.0 40.0 20.0
Initial molar number 1 \vartheta(t) [2,8] [20.0,100.0] 20.0 60.0 40.0 20.0
Initial molar number 1 \vartheta(t) [8,t_{end}] [20.0,100.0] 20.0 60.0 40.0 20.0

Measurement grid

\begin{array}{llll} t_0 = 0 & & & \\ t_{end} = 20 & & & \\ t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20. \end{array}

Source Code

References

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002
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